<!doctype html>
<html style='font-size:19px !important'>
<head>
<meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'>

<style type='text/css'>html {overflow-x: initial !important;}:root { --bg-color:#ffffff; --text-color:#333333; --select-text-bg-color:#B5D6FC; --select-text-font-color:auto; --monospace:"Lucida Console",Consolas,"Courier",monospace; --title-bar-height:20px; }
.mac-os-11 { --title-bar-height:28px; }
html { font-size: 14px; background-color: var(--bg-color); color: var(--text-color); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; -webkit-font-smoothing: antialiased; }
body { margin: 0px; padding: 0px; height: auto; inset: 0px; font-size: 1rem; line-height: 1.42857; overflow-x: hidden; background: inherit; tab-size: 4; }
iframe { margin: auto; }
a.url { word-break: break-all; }
a:active, a:hover { outline: 0px; }
.in-text-selection, ::selection { text-shadow: none; background: var(--select-text-bg-color); color: var(--select-text-font-color); }
#write { margin: 0px auto; height: auto; width: inherit; word-break: normal; overflow-wrap: break-word; position: relative; white-space: normal; overflow-x: visible; padding-top: 36px; }
#write.first-line-indent p { text-indent: 2em; }
#write.first-line-indent li p, #write.first-line-indent p * { text-indent: 0px; }
#write.first-line-indent li { margin-left: 2em; }
.for-image #write { padding-left: 8px; padding-right: 8px; }
body.typora-export { padding-left: 30px; padding-right: 30px; }
.typora-export .footnote-line, .typora-export li, .typora-export p { white-space: pre-wrap; }
.typora-export .task-list-item input { pointer-events: none; }
@media screen and (max-width: 500px) {
  body.typora-export { padding-left: 0px; padding-right: 0px; }
  #write { padding-left: 20px; padding-right: 20px; }
  .CodeMirror-sizer { margin-left: 0px !important; }
  .CodeMirror-gutters { display: none !important; }
}
#write li > figure:last-child { margin-bottom: 0.5rem; }
#write ol, #write ul { position: relative; }
img { max-width: 100%; vertical-align: middle; image-orientation: from-image; }
button, input, select, textarea { color: inherit; font: inherit; }
input[type="checkbox"], input[type="radio"] { line-height: normal; padding: 0px; }
*, ::after, ::before { box-sizing: border-box; }
#write h1, #write h2, #write h3, #write h4, #write h5, #write h6, #write p, #write pre { width: inherit; }
#write h1, #write h2, #write h3, #write h4, #write h5, #write h6, #write p { position: relative; }
p { line-height: inherit; }
h1, h2, h3, h4, h5, h6 { break-after: avoid-page; break-inside: avoid; orphans: 4; }
p { orphans: 4; }
h1 { font-size: 2rem; }
h2 { font-size: 1.8rem; }
h3 { font-size: 1.6rem; }
h4 { font-size: 1.4rem; }
h5 { font-size: 1.2rem; }
h6 { font-size: 1rem; }
.md-math-block, .md-rawblock, h1, h2, h3, h4, h5, h6, p { margin-top: 1rem; margin-bottom: 1rem; }
.hidden { display: none; }
.md-blockmeta { color: rgb(204, 204, 204); font-weight: 700; font-style: italic; }
a { cursor: pointer; }
sup.md-footnote { padding: 2px 4px; background-color: rgba(238, 238, 238, 0.7); color: rgb(85, 85, 85); border-radius: 4px; cursor: pointer; }
sup.md-footnote a, sup.md-footnote a:hover { color: inherit; text-transform: inherit; text-decoration: inherit; }
#write input[type="checkbox"] { cursor: pointer; width: inherit; height: inherit; }
figure { overflow-x: auto; margin: 1.2em 0px; max-width: calc(100% + 16px); padding: 0px; }
figure > table { margin: 0px; }
tr { break-inside: avoid; break-after: auto; }
thead { display: table-header-group; }
table { border-collapse: collapse; border-spacing: 0px; width: 100%; overflow: auto; break-inside: auto; text-align: left; }
table.md-table td { min-width: 32px; }
.CodeMirror-gutters { border-right: 0px; background-color: inherit; }
.CodeMirror-linenumber { user-select: none; }
.CodeMirror { text-align: left; }
.CodeMirror-placeholder { opacity: 0.3; }
.CodeMirror pre { padding: 0px 4px; }
.CodeMirror-lines { padding: 0px; }
div.hr:focus { cursor: none; }
#write pre { white-space: pre-wrap; }
#write.fences-no-line-wrapping pre { white-space: pre; }
#write pre.ty-contain-cm { white-space: normal; }
.CodeMirror-gutters { margin-right: 4px; }
.md-fences { font-size: 0.9rem; display: block; break-inside: avoid; text-align: left; overflow: visible; white-space: pre; background: inherit; position: relative !important; }
.md-fences-adv-panel { width: 100%; margin-top: 10px; text-align: center; padding-top: 0px; padding-bottom: 8px; overflow-x: auto; }
#write .md-fences.mock-cm { white-space: pre-wrap; }
.md-fences.md-fences-with-lineno { padding-left: 0px; }
#write.fences-no-line-wrapping .md-fences.mock-cm { white-space: pre; overflow-x: auto; }
.md-fences.mock-cm.md-fences-with-lineno { padding-left: 8px; }
.CodeMirror-line, twitterwidget { break-inside: avoid; }
.footnotes { opacity: 0.8; font-size: 0.9rem; margin-top: 1em; margin-bottom: 1em; }
.footnotes + .footnotes { margin-top: 0px; }
.md-reset { margin: 0px; padding: 0px; border: 0px; outline: 0px; vertical-align: top; background: 0px 0px; text-decoration: none; text-shadow: none; float: none; position: static; width: auto; height: auto; white-space: nowrap; cursor: inherit; -webkit-tap-highlight-color: transparent; line-height: normal; font-weight: 400; text-align: left; box-sizing: content-box; direction: ltr; }
li div { padding-top: 0px; }
blockquote { margin: 1rem 0px; }
li .mathjax-block, li p { margin: 0.5rem 0px; }
li blockquote { margin: 1rem 0px; }
li { margin: 0px; position: relative; }
blockquote > :last-child { margin-bottom: 0px; }
blockquote > :first-child, li > :first-child { margin-top: 0px; }
.footnotes-area { color: rgb(136, 136, 136); margin-top: 0.714rem; padding-bottom: 0.143rem; white-space: normal; }
#write .footnote-line { white-space: pre-wrap; }
@media print {
  body, html { border: 1px solid transparent; height: 99%; break-after: avoid; break-before: avoid; font-variant-ligatures: no-common-ligatures; }
  #write { margin-top: 0px; padding-top: 0px; border-color: transparent !important; }
  .typora-export * { -webkit-print-color-adjust: exact; }
  .typora-export #write { break-after: avoid; }
  .typora-export #write::after { height: 0px; }
  .is-mac table { break-inside: avoid; }
  .typora-export-show-outline .typora-export-sidebar { display: none; }
}
.footnote-line { margin-top: 0.714em; font-size: 0.7em; }
a img, img a { cursor: pointer; }
pre.md-meta-block { font-size: 0.8rem; min-height: 0.8rem; white-space: pre-wrap; background: rgb(204, 204, 204); display: block; overflow-x: hidden; }
p > .md-image:only-child:not(.md-img-error) img, p > img:only-child { display: block; margin: auto; }
#write.first-line-indent p > .md-image:only-child:not(.md-img-error) img { left: -2em; position: relative; }
p > .md-image:only-child { display: inline-block; width: 100%; }
#write .MathJax_Display { margin: 0.8em 0px 0px; }
.md-math-block { width: 100%; }
.md-math-block:not(:empty)::after { display: none; }
.MathJax_ref { fill: currentcolor; }
[contenteditable="true"]:active, [contenteditable="true"]:focus, [contenteditable="false"]:active, [contenteditable="false"]:focus { outline: 0px; box-shadow: none; }
.md-task-list-item { position: relative; list-style-type: none; }
.task-list-item.md-task-list-item { padding-left: 0px; }
.md-task-list-item > input { position: absolute; top: 0px; left: 0px; margin-left: -1.2em; margin-top: calc(1em - 10px); border: none; }
.math { font-size: 1rem; }
.md-toc { min-height: 3.58rem; position: relative; font-size: 0.9rem; border-radius: 10px; }
.md-toc-content { position: relative; margin-left: 0px; }
.md-toc-content::after, .md-toc::after { display: none; }
.md-toc-item { display: block; color: rgb(65, 131, 196); }
.md-toc-item a { text-decoration: none; }
.md-toc-inner:hover { text-decoration: underline; }
.md-toc-inner { display: inline-block; cursor: pointer; }
.md-toc-h1 .md-toc-inner { margin-left: 0px; font-weight: 700; }
.md-toc-h2 .md-toc-inner { margin-left: 2em; }
.md-toc-h3 .md-toc-inner { margin-left: 4em; }
.md-toc-h4 .md-toc-inner { margin-left: 6em; }
.md-toc-h5 .md-toc-inner { margin-left: 8em; }
.md-toc-h6 .md-toc-inner { margin-left: 10em; }
@media screen and (max-width: 48em) {
  .md-toc-h3 .md-toc-inner { margin-left: 3.5em; }
  .md-toc-h4 .md-toc-inner { margin-left: 5em; }
  .md-toc-h5 .md-toc-inner { margin-left: 6.5em; }
  .md-toc-h6 .md-toc-inner { margin-left: 8em; }
}
a.md-toc-inner { font-size: inherit; font-style: inherit; font-weight: inherit; line-height: inherit; }
.footnote-line a:not(.reversefootnote) { color: inherit; }
.md-attr { display: none; }
.md-fn-count::after { content: "."; }
code, pre, samp, tt { font-family: var(--monospace); }
kbd { margin: 0px 0.1em; padding: 0.1em 0.6em; font-size: 0.8em; color: rgb(36, 39, 41); background: rgb(255, 255, 255); border: 1px solid rgb(173, 179, 185); border-radius: 3px; box-shadow: rgba(12, 13, 14, 0.2) 0px 1px 0px, rgb(255, 255, 255) 0px 0px 0px 2px inset; white-space: nowrap; vertical-align: middle; }
.md-comment { color: rgb(162, 127, 3); opacity: 0.8; font-family: var(--monospace); }
code { text-align: left; vertical-align: initial; }
a.md-print-anchor { white-space: pre !important; border-width: initial !important; border-style: none !important; border-color: initial !important; display: inline-block !important; position: absolute !important; width: 1px !important; right: 0px !important; outline: 0px !important; background: 0px 0px !important; text-decoration: initial !important; text-shadow: initial !important; }
.os-windows.monocolor-emoji .md-emoji { font-family: "Segoe UI Symbol", sans-serif; }
.md-diagram-panel > svg { max-width: 100%; }
[lang="flow"] svg, [lang="mermaid"] svg { max-width: 100%; height: auto; }
[lang="mermaid"] .node text { font-size: 1rem; }
table tr th { border-bottom: 0px; }
video { max-width: 100%; display: block; margin: 0px auto; }
iframe { max-width: 100%; width: 100%; border: none; }
.highlight td, .highlight tr { border: 0px; }
mark { background: rgb(255, 255, 0); color: rgb(0, 0, 0); }
.md-html-inline .md-plain, .md-html-inline strong, mark .md-inline-math, mark strong { color: inherit; }
.md-expand mark .md-meta { opacity: 0.3 !important; }
mark .md-meta { color: rgb(0, 0, 0); }
@media print {
  .typora-export h1, .typora-export h2, .typora-export h3, .typora-export h4, .typora-export h5, .typora-export h6 { break-inside: avoid; }
}
.md-diagram-panel .messageText { stroke: none !important; }
.md-diagram-panel .start-state { fill: var(--node-fill); }
.md-diagram-panel .edgeLabel rect { opacity: 1 !important; }
.md-fences.md-fences-math { font-size: 1em; }
.md-fences-advanced:not(.md-focus) { padding: 0px; white-space: nowrap; border: 0px; }
.md-fences-advanced:not(.md-focus) { background: inherit; }
.typora-export-show-outline .typora-export-content { max-width: 1440px; margin: auto; display: flex; flex-direction: row; }
.typora-export-sidebar { width: 300px; font-size: 0.8rem; margin-top: 80px; margin-right: 18px; }
.typora-export-show-outline #write { --webkit-flex:2; flex: 2 1 0%; }
.typora-export-sidebar .outline-content { position: fixed; top: 0px; max-height: 100%; overflow: hidden auto; padding-bottom: 30px; padding-top: 60px; width: 300px; }
@media screen and (max-width: 1024px) {
  .typora-export-sidebar, .typora-export-sidebar .outline-content { width: 240px; }
}
@media screen and (max-width: 800px) {
  .typora-export-sidebar { display: none; }
}
.outline-content li, .outline-content ul { margin-left: 0px; margin-right: 0px; padding-left: 0px; padding-right: 0px; list-style: none; }
.outline-content ul { margin-top: 0px; margin-bottom: 0px; }
.outline-content strong { font-weight: 400; }
.outline-expander { width: 1rem; height: 1.42857rem; position: relative; display: table-cell; vertical-align: middle; cursor: pointer; padding-left: 4px; }
.outline-expander::before { content: ""; position: relative; font-family: Ionicons; display: inline-block; font-size: 8px; vertical-align: middle; }
.outline-item { padding-top: 3px; padding-bottom: 3px; cursor: pointer; }
.outline-expander:hover::before { content: ""; }
.outline-h1 > .outline-item { padding-left: 0px; }
.outline-h2 > .outline-item { padding-left: 1em; }
.outline-h3 > .outline-item { padding-left: 2em; }
.outline-h4 > .outline-item { padding-left: 3em; }
.outline-h5 > .outline-item { padding-left: 4em; }
.outline-h6 > .outline-item { padding-left: 5em; }
.outline-label { cursor: pointer; display: table-cell; vertical-align: middle; text-decoration: none; color: inherit; }
.outline-label:hover { text-decoration: underline; }
.outline-item:hover { border-color: rgb(245, 245, 245); background-color: var(--item-hover-bg-color); }
.outline-item:hover { margin-left: -28px; margin-right: -28px; border-left: 28px solid transparent; border-right: 28px solid transparent; }
.outline-item-single .outline-expander::before, .outline-item-single .outline-expander:hover::before { display: none; }
.outline-item-open > .outline-item > .outline-expander::before { content: ""; }
.outline-children { display: none; }
.info-panel-tab-wrapper { display: none; }
.outline-item-open > .outline-children { display: block; }
.typora-export .outline-item { padding-top: 1px; padding-bottom: 1px; }
.typora-export .outline-item:hover { margin-right: -8px; border-right: 8px solid transparent; }
.typora-export .outline-expander::before { content: "+"; font-family: inherit; top: -1px; }
.typora-export .outline-expander:hover::before, .typora-export .outline-item-open > .outline-item > .outline-expander::before { content: "−"; }
.typora-export-collapse-outline .outline-children { display: none; }
.typora-export-collapse-outline .outline-item-open > .outline-children, .typora-export-no-collapse-outline .outline-children { display: block; }
.typora-export-no-collapse-outline .outline-expander::before { content: "" !important; }
.typora-export-show-outline .outline-item-active > .outline-item .outline-label { font-weight: 700; }
.md-inline-math-container mjx-container { zoom: 0.95; }


.CodeMirror { height: auto; }
.CodeMirror.cm-s-inner { background: inherit; }
.CodeMirror-scroll { overflow: auto hidden; z-index: 3; }
.CodeMirror-gutter-filler, .CodeMirror-scrollbar-filler { background-color: rgb(255, 255, 255); }
.CodeMirror-gutters { border-right: 1px solid rgb(221, 221, 221); background: inherit; white-space: nowrap; }
.CodeMirror-linenumber { padding: 0px 3px 0px 5px; text-align: right; color: rgb(153, 153, 153); }
.cm-s-inner .cm-keyword { color: rgb(119, 0, 136); }
.cm-s-inner .cm-atom, .cm-s-inner.cm-atom { color: rgb(34, 17, 153); }
.cm-s-inner .cm-number { color: rgb(17, 102, 68); }
.cm-s-inner .cm-def { color: rgb(0, 0, 255); }
.cm-s-inner .cm-variable { color: rgb(0, 0, 0); }
.cm-s-inner .cm-variable-2 { color: rgb(0, 85, 170); }
.cm-s-inner .cm-variable-3 { color: rgb(0, 136, 85); }
.cm-s-inner .cm-string { color: rgb(170, 17, 17); }
.cm-s-inner .cm-property { color: rgb(0, 0, 0); }
.cm-s-inner .cm-operator { color: rgb(152, 26, 26); }
.cm-s-inner .cm-comment, .cm-s-inner.cm-comment { color: rgb(170, 85, 0); }
.cm-s-inner .cm-string-2 { color: rgb(255, 85, 0); }
.cm-s-inner .cm-meta { color: rgb(85, 85, 85); }
.cm-s-inner .cm-qualifier { color: rgb(85, 85, 85); }
.cm-s-inner .cm-builtin { color: rgb(51, 0, 170); }
.cm-s-inner .cm-bracket { color: rgb(153, 153, 119); }
.cm-s-inner .cm-tag { color: rgb(17, 119, 0); }
.cm-s-inner .cm-attribute { color: rgb(0, 0, 204); }
.cm-s-inner .cm-header, .cm-s-inner.cm-header { color: rgb(0, 0, 255); }
.cm-s-inner .cm-quote, .cm-s-inner.cm-quote { color: rgb(0, 153, 0); }
.cm-s-inner .cm-hr, .cm-s-inner.cm-hr { color: rgb(153, 153, 153); }
.cm-s-inner .cm-link, .cm-s-inner.cm-link { color: rgb(0, 0, 204); }
.cm-negative { color: rgb(221, 68, 68); }
.cm-positive { color: rgb(34, 153, 34); }
.cm-header, .cm-strong { font-weight: 700; }
.cm-del { text-decoration: line-through; }
.cm-em { font-style: italic; }
.cm-link { text-decoration: underline; }
.cm-error { color: red; }
.cm-invalidchar { color: red; }
.cm-constant { color: rgb(38, 139, 210); }
.cm-defined { color: rgb(181, 137, 0); }
div.CodeMirror span.CodeMirror-matchingbracket { color: rgb(0, 255, 0); }
div.CodeMirror span.CodeMirror-nonmatchingbracket { color: rgb(255, 34, 34); }
.cm-s-inner .CodeMirror-activeline-background { background: inherit; }
.CodeMirror { position: relative; overflow: hidden; }
.CodeMirror-scroll { height: 100%; outline: 0px; position: relative; box-sizing: content-box; background: inherit; }
.CodeMirror-sizer { position: relative; }
.CodeMirror-gutter-filler, .CodeMirror-hscrollbar, .CodeMirror-scrollbar-filler, .CodeMirror-vscrollbar { position: absolute; z-index: 6; display: none; outline: 0px; }
.CodeMirror-vscrollbar { right: 0px; top: 0px; overflow: hidden; }
.CodeMirror-hscrollbar { bottom: 0px; left: 0px; overflow: auto hidden; }
.CodeMirror-scrollbar-filler { right: 0px; bottom: 0px; }
.CodeMirror-gutter-filler { left: 0px; bottom: 0px; }
.CodeMirror-gutters { position: absolute; left: 0px; top: 0px; padding-bottom: 10px; z-index: 3; overflow-y: hidden; }
.CodeMirror-gutter { white-space: normal; height: 100%; box-sizing: content-box; padding-bottom: 30px; margin-bottom: -32px; display: inline-block; }
.CodeMirror-gutter-wrapper { position: absolute; z-index: 4; background: 0px 0px !important; border: none !important; }
.CodeMirror-gutter-background { position: absolute; top: 0px; bottom: 0px; z-index: 4; }
.CodeMirror-gutter-elt { position: absolute; cursor: default; z-index: 4; }
.CodeMirror-lines { cursor: text; }
.CodeMirror pre { border-radius: 0px; border-width: 0px; background: 0px 0px; font-family: inherit; font-size: inherit; margin: 0px; white-space: pre; overflow-wrap: normal; color: inherit; z-index: 2; position: relative; overflow: visible; }
.CodeMirror-wrap pre { overflow-wrap: break-word; white-space: pre-wrap; word-break: normal; }
.CodeMirror-code pre { border-right: 30px solid transparent; width: fit-content; }
.CodeMirror-wrap .CodeMirror-code pre { border-right: none; width: auto; }
.CodeMirror-linebackground { position: absolute; inset: 0px; z-index: 0; }
.CodeMirror-linewidget { position: relative; z-index: 2; overflow: auto; }
.CodeMirror-wrap .CodeMirror-scroll { overflow-x: hidden; }
.CodeMirror-measure { position: absolute; width: 100%; height: 0px; overflow: hidden; visibility: hidden; }
.CodeMirror-measure pre { position: static; }
.CodeMirror div.CodeMirror-cursor { position: absolute; visibility: hidden; border-right: none; width: 0px; }
.CodeMirror div.CodeMirror-cursor { visibility: hidden; }
.CodeMirror-focused div.CodeMirror-cursor { visibility: inherit; }
.cm-searching { background: rgba(255, 255, 0, 0.4); }
span.cm-underlined { text-decoration: underline; }
span.cm-strikethrough { text-decoration: line-through; }
.cm-tw-syntaxerror { color: rgb(255, 255, 255); background-color: rgb(153, 0, 0); }
.cm-tw-deleted { text-decoration: line-through; }
.cm-tw-header5 { font-weight: 700; }
.cm-tw-listitem:first-child { padding-left: 10px; }
.cm-tw-box { border-style: solid; border-right-width: 1px; border-bottom-width: 1px; border-left-width: 1px; border-color: inherit; border-top-width: 0px !important; }
.cm-tw-underline { text-decoration: underline; }
@media print {
  .CodeMirror div.CodeMirror-cursor { visibility: hidden; }
}


 :root {
    --side-bar-bg-color: #d9ede5;
	--control-text-color: #6B6B6B;
	--active-file-bg-color: #ecf6f2;
	--active-file-border-color: #6B6B6B;
	--active-file-text-color: #202020;
	--table-even-row-color:#f8fcfa;
	--table-head-color:#d9ede5;
	--deep-theme-color: #4eb289;
	--code-block-bg-color: #0F111A;

	
}

/*serif*/
/*monospace*/
/*Chinese*/
html{
	font-size: 16px;
}

body {
    font-family: 'Lexend','SourceCodePro','NotoSansSC';
	font-weight: normal;
	line-height: 1.5rem;
	letter-spacing: 0;
    margin: 0;
}

#write {
	max-width: 900px;
    padding: 30px 50px 20px;
}

#write p{
	text-align:left;
}

#write pre.md-meta-block {
	padding: 1rem;
	font-size: 85%;
	line-height: 1.45;
	background-color: #f7f7f7;
	border: 0;
	border-radius: 3px;
	color: #777777;
	margin-top: 0 !important;
}

.md-image>.md-meta {
    color: #777777;
    font-size: 0.9rem;
	font-family: 'Lexend';
    padding: 4px 0;
}


@media print {
  html {
    font-size: 12px;
  }

  table,
  pre {
    page-break-inside: avoid;
  }

  pre {
    word-wrap: break-word;
  }
  
  @page {
    size: A4; 
    margin-left: 0;
    margin-right: 0;
  }
}

/*toc*/
.md-toc { 
    margin-top:20px;
    padding-bottom:20px;
	color: var(--deep-theme-color);
}

a {
	color: var(--deep-theme-color);
	text-decoration: none;
}

a:hover {
    text-decoration: underline;
}

/*headers*/
h1,h2,h3,h4,h5,h6 {
	display: block;
	font-weight:bold;
}

h1 {
	font-size: 2em;
	margin-top: 0.67em;
	margin-bottom: 0.67em;
}

h2 {
	font-size: 1.5em;
	margin-top: 0.83em;
	margin-bottom: 0.83em;
}

h3 {
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<body class='typora-export typora-export-show-outline typora-export-collapse-outline'><div class='typora-export-content'>
<div class="typora-export-sidebar"><div class="outline-content"><li class="outline-item-wrapper outline-h1 outline-item-open"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#线性方程的迭代方法">线性方程的迭代方法</a></div><ul class="outline-children"><li class="outline-item-wrapper outline-h2 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#静态stationary）方法">静态（Stationary）方法</a></div><ul class="outline-children"></ul></li><li class="outline-item-wrapper outline-h2 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#共轭梯度法conjugate-gradient-cg">共轭梯度法(Conjugate gradient, CG)</a></div><ul class="outline-children"></ul></li><li class="outline-item-wrapper outline-h2 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#bicg">BiCG</a></div><ul class="outline-children"></ul></li></ul></li></div></div><div id='write'  class=''><h1 id='线性方程的迭代方法'><span>线性方程的迭代方法</span></h1><p><span>本文将介绍求解线性方程 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.979ex" height="1.805ex" role="img" focusable="false" viewBox="0 -716 3084.6 798" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path id="MJX-134-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-134-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-134-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-134-TEX-I-1D44F" d="M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 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xlink:href="#MJX-134-TEX-I-1D44F"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>x</mi><mo>=</mo><mi>b</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">Ax = b</script><span> 的几种方法。  其中 A 是一个大型稀疏矩阵，通常以算子的形式给出，例如在偏微分方程中。这类问题的规模太大，以至于像LU分解之类的直接方法受内存限制无法使用，故需要使用迭代求解。</span></p><h2 id='静态stationary）方法'><span>静态（Stationary）方法</span></h2><p><span>采用如下迭代格式： </span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n5" cid="n5" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width: 42.657ex; position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="2.565ex" role="img" focusable="false" 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xlink:href="#MJX-110-TEX-I-1D434"></use></g><g data-mml-node="mo" transform="translate(7491.6,0)"><use data-c="2212" xlink:href="#MJX-110-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(8491.8,0)"><use data-c="1D440" xlink:href="#MJX-110-TEX-I-1D440"></use></g><g data-mml-node="mo" transform="translate(9542.8,0)"><use data-c="29" xlink:href="#MJX-110-TEX-N-29"></use></g><g data-mml-node="msub" transform="translate(9931.8,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-110-TEX-I-1D465"></use></g><g data-mml-node="mi" transform="translate(605,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-110-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(11177.4,0)"><use data-c="2B" xlink:href="#MJX-110-TEX-N-2B"></use></g><g data-mml-node="msup" transform="translate(12177.7,0)"><g data-mml-node="mi"><use data-c="1D440" xlink:href="#MJX-110-TEX-I-1D440"></use></g><g data-mml-node="TeXAtom" transform="translate(1138,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-110-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="31" xlink:href="#MJX-110-TEX-N-31"></use></g></g></g><g data-mml-node="mi" transform="translate(14269.4,0)"><use data-c="1D44F" xlink:href="#MJX-110-TEX-I-1D44F"></use></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -817 1 1133.9"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-mjx-eqn:1" transform="translate(0,683)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-750)"><g data-mml-node="mtext"><use data-c="28" xlink:href="#MJX-110-TEX-N-28"></use><use data-c="31" xlink:href="#MJX-110-TEX-N-31" transform="translate(389,0)"></use><use data-c="29" xlink:href="#MJX-110-TEX-N-29" transform="translate(889,0)"></use></g></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(1)</mtext></mtd><mtd><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mo>−</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mi>M</mi><mo stretchy="false">)</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup><mi>b</mi></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>M是一个相对于A更便于求逆的矩阵。可以很方便的验证上述格式在</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.979ex" height="1.805ex" role="img" focusable="false" viewBox="0 -716 3084.6 798" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path 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54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-135-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-135-TEX-I-1D44F" d="M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D434" xlink:href="#MJX-135-TEX-I-1D434"></use></g><g data-mml-node="mi" transform="translate(750,0)"><use data-c="1D465" xlink:href="#MJX-135-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1599.8,0)"><use data-c="3D" xlink:href="#MJX-135-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(2655.6,0)"><use data-c="1D44F" xlink:href="#MJX-135-TEX-I-1D44F"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>x</mi><mo>=</mo><mi>b</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">Ax=b</script><span> 是不动点。M有3种常用的选择</span></p><ol start='' ><li><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-141-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D434" xlink:href="#MJX-141-TEX-I-1D434"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">A</script><span> 的对角元 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.873ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 828 683" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-137-TEX-I-1D437" d="M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z"></path></defs><g 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">A</script><span> 的下三角元 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.18ex" height="1.731ex" role="img" focusable="false" viewBox="0 -683 2731.4 765" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path id="MJX-139-TEX-I-1D437" d="M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z"></path><path 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stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mi>ω</mi><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>i</mi></mrow></msub></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mi>b</mi><mi>i</mi></msub><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow><mi>ω</mi><mo>−</mo><mn>1</mn></mrow><mi>ω</mi></mfrac></mrow><msubsup><mi>x</mi><mi>i</mi><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>−</mo><munder><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>j</mi><mo>&lt;</mo><mi>i</mi></mrow></munder><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub><msubsup><mi>x</mi><mi>j</mi><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>−</mo><munder><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>j</mi><mo>&gt;</mo><mi>i</mi></mrow></munder><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub><msubsup><mi>x</mi><mi>j</mi><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msubsup><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><h2 id='共轭梯度法conjugate-gradient-cg'><span>共轭梯度法(Conjugate gradient, CG)</span></h2><p><span>当</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-141-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 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transform="translate(220,-727.7)"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D45D" xlink:href="#MJX-114-TEX-I-1D45D"></use></g><g data-mml-node="mi" transform="translate(536,289) scale(0.707)"><use data-c="1D447" xlink:href="#MJX-114-TEX-I-1D447"></use></g></g><g data-mml-node="mi" transform="translate(1083.8,0)"><use data-c="1D434" xlink:href="#MJX-114-TEX-I-1D434"></use></g><g data-mml-node="mi" transform="translate(1833.8,0)"><use data-c="1D45D" xlink:href="#MJX-114-TEX-I-1D45D"></use></g></g><rect width="2536.8" height="60" x="120" y="220"></rect></g></g><g data-mml-node="mo" transform="translate(13565.7,0)"><use data-c="2C" xlink:href="#MJX-114-TEX-N-2C"></use></g><g data-mml-node="mstyle" transform="translate(13843.7,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mtext" transform="translate(15010.4,0)"><use data-c="77" xlink:href="#MJX-114-TEX-N-77"></use><use data-c="68" xlink:href="#MJX-114-TEX-N-68" transform="translate(722,0)"></use><use data-c="65" xlink:href="#MJX-114-TEX-N-65" transform="translate(1278,0)"></use><use data-c="72" xlink:href="#MJX-114-TEX-N-72" transform="translate(1722,0)"></use><use data-c="65" xlink:href="#MJX-114-TEX-N-65" transform="translate(2114,0)"></use><use data-c="A0" xlink:href="#MJX-114-TEX-N-A0" transform="translate(2558,0)"></use></g><g data-mml-node="mi" transform="translate(17818.4,0)"><use data-c="1D45F" xlink:href="#MJX-114-TEX-I-1D45F"></use></g><g data-mml-node="mo" transform="translate(18547.1,0)"><use data-c="3D" xlink:href="#MJX-114-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(19602.9,0)"><use data-c="1D44F" xlink:href="#MJX-114-TEX-I-1D44F"></use></g><g data-mml-node="mo" transform="translate(20254.1,0)"><use data-c="2212" xlink:href="#MJX-114-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(21254.4,0)"><use data-c="1D434" xlink:href="#MJX-114-TEX-I-1D434"></use></g><g data-mml-node="mi" transform="translate(22004.4,0)"><use data-c="1D465" xlink:href="#MJX-114-TEX-I-1D465"></use></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -1469.7 1 2439.4"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-mjx-eqn:5" transform="translate(0,702)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-750)"><g data-mml-node="mtext"><use data-c="28" xlink:href="#MJX-114-TEX-N-28"></use><use data-c="35" xlink:href="#MJX-114-TEX-N-35" transform="translate(389,0)"></use><use data-c="29" xlink:href="#MJX-114-TEX-N-29" transform="translate(889,0)"></use></g></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(5)</mtext></mtd><mtd><mi>a</mi><mo>=</mo><mi>arg</mi><mo data-mjx-texclass="NONE">⁡</mo><mo data-mjx-texclass="OP" movablelimits="true">min</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>a</mi><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow><msup><mi>p</mi><mi>T</mi></msup><mi>r</mi></mrow><mrow><msup><mi>p</mi><mi>T</mi></msup><mi>A</mi><mi>p</mi></mrow></mfrac></mrow><mo>,</mo><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><mtext>where&nbsp;</mtext><mi>r</mi><mo>=</mo><mi>b</mi><mo>−</mo><mi>A</mi><mi>x</mi></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>再把 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.395ex" height="1.758ex" role="img" focusable="false" viewBox="0 -583 2826.4 777" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-146-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-146-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-146-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 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393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-146-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(794.2,0)"><use data-c="2B" xlink:href="#MJX-146-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(1794.4,0)"><use data-c="1D44E" xlink:href="#MJX-146-TEX-I-1D44E"></use></g><g data-mml-node="mi" transform="translate(2323.4,0)"><use data-c="1D45D" xlink:href="#MJX-146-TEX-I-1D45D"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>a</mi><mi>p</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">x+ap</script><span> 当做新的起点，换个方向重复以上步骤就可以不断减小 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.02ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 451 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-151-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-151-TEX-I-1D45F"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">r</script><span> 直到找到解。剩下就是选择搜索的方向。注意到</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="12.085ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 5341.6 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-148-TEX-N-2207" d="M46 676Q46 679 51 683H781Q786 679 786 676Q786 674 617 326T444 -26Q439 -33 416 -33T388 -26Q385 -22 216 326T46 676ZM697 596Q697 597 445 597T193 596Q195 591 319 336T445 80L697 596Z"></path><path id="MJX-148-TEX-I-1D719" d="M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 418 563Q411 535 404 504T392 458L388 442Q388 441 397 441T429 435T477 418Q521 397 550 357T579 260T548 151T471 65T374 11T279 -10H275L251 -105Q245 -128 238 -160Q230 -192 227 -198T215 -205H209Q189 -205 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transform="translate(1818,0)"><use data-c="1D465" xlink:href="#MJX-148-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(2390,0)"><use data-c="29" xlink:href="#MJX-148-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(3056.8,0)"><use data-c="3D" xlink:href="#MJX-148-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(4112.6,0)"><use data-c="2212" xlink:href="#MJX-148-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(4890.6,0)"><use data-c="1D45F" xlink:href="#MJX-148-TEX-I-1D45F"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">∇</mi><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mi>r</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\nabla\phi(x)=-r</script><span>, 即 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.02ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 451 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-151-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-151-TEX-I-1D45F"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">r</script><span> 是 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.403ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1946 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-150-TEX-I-1D719" d="M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 418 563Q411 535 404 504T392 458L388 442Q388 441 397 441T429 435T477 418Q521 397 550 357T579 260T548 151T471 65T374 11T279 -10H275L251 -105Q245 -128 238 -160Q230 -192 227 -198T215 -205H209Q189 -205 189 -198Q189 -193 211 -103L234 -11Q234 -10 226 -10Q221 -10 206 -8T161 6T107 36T62 89T43 171Q43 231 76 284T157 370T254 422T342 441Q347 441 348 445L378 567Q409 686 409 688ZM122 150Q122 116 134 91T167 53T203 35T237 27H244L337 404Q333 404 326 403T297 395T255 379T211 350T170 304Q152 276 137 237Q122 191 122 150ZM500 282Q500 320 484 347T444 385T405 400T381 404H378L332 217L284 29Q284 27 285 27Q293 27 317 33T357 47Q400 66 431 100T475 170T494 234T500 282Z"></path><path id="MJX-150-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-150-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-150-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D719" xlink:href="#MJX-150-TEX-I-1D719"></use></g><g data-mml-node="mo" transform="translate(596,0)"><use data-c="28" xlink:href="#MJX-150-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(985,0)"><use data-c="1D465" xlink:href="#MJX-150-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1557,0)"><use data-c="29" xlink:href="#MJX-150-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">\phi(x)</script><span> 下降最快的方向，故可以选取 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.02ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 451 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-151-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-151-TEX-I-1D45F"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">r</script><span> 作为每次搜索的方向。但实际上有更好的办法。记 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="26.944ex" height="2.444ex" role="img" focusable="false" viewBox="0 -830.4 11909.3 1080.4" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-152-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 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transform="translate(1123,0)"><use data-c="31" xlink:href="#MJX-152-TEX-N-31"></use></g></g></g><g data-mml-node="mo" transform="translate(8385.4,0)"><use data-c="5D" xlink:href="#MJX-152-TEX-N-5D"></use></g><g data-mml-node="mo" transform="translate(8941.2,0)"><use data-c="2208" xlink:href="#MJX-152-TEX-N-2208"></use></g><g data-mml-node="msup" transform="translate(9885.9,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="211D" xlink:href="#MJX-152-TEX-D-211D"></use></g></g><g data-mml-node="TeXAtom" transform="translate(755,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-152-TEX-I-1D45B"></use></g><g data-mml-node="mi" transform="translate(600,0)"><use data-c="D7" xlink:href="#MJX-152-TEX-I-D7"></use></g><g data-mml-node="mi" transform="translate(1378,0)"><use data-c="1D456" xlink:href="#MJX-152-TEX-I-1D456"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><mo stretchy="false">[</mo><msub><mi>p</mi><mn>0</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">]</mo><mo>∈</mo><msup><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">R</mi></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi><mi>×</mi><mi>i</mi></mrow></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex">P_{i-1}=[p_0, \cdots, p_{i-1}]\in\mathbb{R}^{n\cross i}</script><span> 为前 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.781ex" height="1.52ex" role="img" focusable="false" viewBox="0 -661 345 672" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-157-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-157-TEX-I-1D456"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">i</script><span> 次的搜索方向，则 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xlink:href="#MJX-118-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1123,0)"><use data-c="31" xlink:href="#MJX-118-TEX-N-31"></use></g></g></g></g></g><g data-mml-node="mtext" transform="translate(5780.9,0)"><use data-c="A0" xlink:href="#MJX-118-TEX-N-A0"></use></g><g data-mml-node="mi" transform="translate(6030.9,0)"><use data-c="1D719" xlink:href="#MJX-118-TEX-I-1D719"></use></g><g data-mml-node="mo" transform="translate(6626.9,0)"><use data-c="28" xlink:href="#MJX-118-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(7015.9,0)"><use data-c="1D465" xlink:href="#MJX-118-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(7587.9,0)"><use data-c="29" xlink:href="#MJX-118-TEX-N-29"></use></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -1184.3 1 1868.6"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-mjx-eqn:9" transform="translate(0,1184.3)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-750)"><g data-mml-node="mtext"><use data-c="28" xlink:href="#MJX-118-TEX-N-28"></use><use data-c="39" xlink:href="#MJX-118-TEX-N-39" transform="translate(389,0)"></use><use data-c="29" xlink:href="#MJX-118-TEX-N-29" transform="translate(889,0)"></use></g></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(9)</mtext></mtd><mtd><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><munder><mrow><mi>arg</mi><mo data-mjx-texclass="NONE">⁡</mo><mo data-mjx-texclass="OP" movablelimits="true">min</mo></mrow><mrow><mi>x</mi><mo>∈</mo><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><msub><mrow data-mjx-texclass="ORD"><mi data-mjx-variant="-tex-calligraphic" mathvariant="script">K</mi></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></munder><mtext>&nbsp;</mtext><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>由此，我们选择 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.878ex" height="1.439ex" role="img" focusable="false" viewBox="0 -442 830 636" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-170-TEX-I-1D45D" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path id="MJX-170-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45D" xlink:href="#MJX-170-TEX-I-1D45D"></use></g><g data-mml-node="mi" transform="translate(536,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-170-TEX-I-1D456"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">p_i</script><span> 为 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.76ex" height="1.357ex" role="img" focusable="false" viewBox="0 -442 778 599.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-171-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-171-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g 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-1 0 0)"><g data-mml-node="mtd" id="mjx-mjx-eqn:10" transform="translate(0,682.3)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-750)"><g data-mml-node="mtext"><use data-c="28" xlink:href="#MJX-119-TEX-N-28"></use><use data-c="31" xlink:href="#MJX-119-TEX-N-31" transform="translate(389,0)"></use><use data-c="30" xlink:href="#MJX-119-TEX-N-30" transform="translate(889,0)"></use><use data-c="29" xlink:href="#MJX-119-TEX-N-29" transform="translate(1389,0)"></use></g></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(10)</mtext></mtd><mtd><msub><mi>r</mi><mi>i</mi></msub><mo>=</mo><msub><mi>γ</mi><mi>i</mi></msub><msub><mi>p</mi><mi>i</mi></msub><mo>+</mo><msub><mi>z</mi><mi>i</mi></msub><mo>,</mo><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><mtext>where&nbsp;</mtext><msub><mi>z</mi><mi>i</mi></msub><mo>∈</mo><mi>A</mi><msub><mrow data-mjx-texclass="ORD"><mi data-mjx-variant="-tex-calligraphic" mathvariant="script">K</mi></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mtext>&nbsp;</mtext><msub><mi>p</mi><mi>i</mi></msub><mo>∈</mo><mo stretchy="false">(</mo><mi>A</mi><msub><mrow data-mjx-texclass="ORD"><mi data-mjx-variant="-tex-calligraphic" mathvariant="script">K</mi></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msup><mo stretchy="false">)</mo><mo>⊥</mo></msup></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>到此我们已经得到了一个算法。但其在每步中需要使用 Gram-Schmidt 方法来求 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.878ex" height="1.439ex" role="img" focusable="false" viewBox="0 -442 830 636" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-170-TEX-I-1D45D" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path id="MJX-170-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45D" xlink:href="#MJX-170-TEX-I-1D45D"></use></g><g data-mml-node="mi" transform="translate(536,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-170-TEX-I-1D456"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">p_i</script><span> 。下面说明 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.878ex" height="1.439ex" role="img" focusable="false" viewBox="0 -442 830 636" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-170-TEX-I-1D45D" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path id="MJX-170-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45D" xlink:href="#MJX-170-TEX-I-1D45D"></use></g><g data-mml-node="mi" transform="translate(536,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-170-TEX-I-1D456"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">p_i</script><span> 是 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.76ex" height="1.357ex" role="img" focusable="false" viewBox="0 -442 778 599.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-171-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-171-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-171-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(484,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-171-TEX-I-1D456"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">r_i</script><span> 和 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.922ex" height="1.471ex" role="img" focusable="false" viewBox="0 -442 1733.6 650" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.471ex;"><defs><path id="MJX-172-TEX-I-1D45D" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path id="MJX-172-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path id="MJX-172-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-172-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45D" xlink:href="#MJX-172-TEX-I-1D45D"></use></g><g data-mml-node="TeXAtom" transform="translate(536,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-172-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(345,0)"><use data-c="2212" xlink:href="#MJX-172-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1123,0)"><use data-c="31" xlink:href="#MJX-172-TEX-N-31"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">p_{i-1}</script><span> 的线性组合，从而对算法优化。先证几个命题。</span></p><p><strong><span>Proposition 1:</span></strong><span>  </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="24.384ex" height="2.723ex" role="img" focusable="false" viewBox="0 -853.7 10777.7 1203.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.792ex;"><defs><path id="MJX-173-TEX-C-4B" d="M194 618Q193 618 182 613T156 601T131 594Q113 594 113 605Q113 623 144 644Q154 650 205 676T267 703Q277 705 279 705Q295 705 295 691Q295 569 250 397Q225 306 197 217T151 81T128 25Q120 8 94 -7T47 -22Q32 -22 32 -10L64 76Q95 163 133 295T185 530Q198 611 194 618ZM331 429Q331 383 364 290T449 117T542 36Q574 36 607 51T652 103Q660 124 677 133T709 143Q727 143 727 128Q727 119 723 111Q704 56 639 17T497 -22H493Q463 -22 425 16Q401 40 382 71Q335 138 296 243T256 399Q256 434 288 473Q342 540 471 622T670 705Q691 704 703 696Q732 678 732 644Q732 613 714 600T677 586Q671 586 667 587T660 592T657 604V619Q657 647 629 647Q623 647 620 646Q576 635 495 583T365 482Q331 448 331 429Z"></path><path id="MJX-173-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path id="MJX-173-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-173-TEX-SO-27E8" d="M373 850Q392 850 394 832Q394 825 267 538L139 250L267 -38Q394 -325 394 -332Q392 -350 375 -350Q361 -350 356 -338Q354 -331 289 -186T161 103T97 250T160 397T289 685T356 838Q362 850 373 850Z"></path><path id="MJX-173-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-173-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-173-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-173-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-173-TEX-N-22EF" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250ZM525 250Q525 274 542 292T585 310Q609 310 627 294T646 251Q646 226 629 208T586 190T543 207T525 250ZM972 250Q972 274 989 292T1032 310Q1056 310 1074 294T1093 251Q1093 226 1076 208T1033 190T990 207T972 250Z"></path><path id="MJX-173-TEX-SO-27E9" d="M77 832Q77 837 82 843T98 850Q110 849 115 838Q117 831 182 686T310 397T374 250T311 103T182 -185T115 -338Q110 -350 96 -350Q79 -350 77 -332Q77 -325 204 -38L332 250L204 538Q77 825 77 832Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="4B" xlink:href="#MJX-173-TEX-C-4B"></use></g></g><g data-mml-node="mi" transform="translate(795,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-173-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(1491.2,0)"><use data-c="3D" xlink:href="#MJX-173-TEX-N-3D"></use></g><g data-mml-node="mrow" transform="translate(2547,0)"><g data-mml-node="mo"><use data-c="27E8" xlink:href="#MJX-173-TEX-SO-27E8"></use></g><g data-mml-node="msub" transform="translate(472,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-173-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-173-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(1359.6,0)"><use data-c="2C" xlink:href="#MJX-173-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(1804.2,0)"><use data-c="1D434" xlink:href="#MJX-173-TEX-I-1D434"></use></g><g data-mml-node="msub" transform="translate(2554.2,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-173-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-173-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(3441.8,0)"><use data-c="2C" xlink:href="#MJX-173-TEX-N-2C"></use></g><g data-mml-node="mo" transform="translate(3886.4,0)"><use data-c="22EF" xlink:href="#MJX-173-TEX-N-22EF"></use></g><g data-mml-node="mo" transform="translate(5225.1,0)"><use data-c="2C" xlink:href="#MJX-173-TEX-N-2C"></use></g><g data-mml-node="msup" transform="translate(5669.8,0)"><g data-mml-node="mi"><use data-c="1D434" xlink:href="#MJX-173-TEX-I-1D434"></use></g><g data-mml-node="mi" transform="translate(783,363) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-173-TEX-I-1D458"></use></g></g><g data-mml-node="msub" transform="translate(6871.2,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-173-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-173-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(7758.7,0)"><use data-c="27E9" xlink:href="#MJX-173-TEX-SO-27E9"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi data-mjx-variant="-tex-calligraphic" mathvariant="script">K</mi></mrow><mi>k</mi></msub><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>r</mi><mn>0</mn></msub><mo>,</mo><mi>A</mi><msub><mi>r</mi><mn>0</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msup><mi>A</mi><mi>k</mi></msup><msub><mi>r</mi><mn>0</mn></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\mathcal{K}_k=\left<r_0,Ar_0,\cdots,A^kr_0\right></script></p><p><strong><span>Proof: </span></strong><span> 使用归纳法。</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.327ex" height="1.756ex" role="img" focusable="false" viewBox="0 -694 2354.6 776" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path id="MJX-174-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path id="MJX-174-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-174-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-174-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(798.8,0)"><use data-c="3D" xlink:href="#MJX-174-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1854.6,0)"><use data-c="30" xlink:href="#MJX-174-TEX-N-30"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">k=0</script><span> 时显然成立，设命题对于 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.179ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 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stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-175-TEX-I-1D458"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">k</script><span> 成立，注意到 </span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n35" cid="n35" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width: 60.117ex; position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="2.784ex" role="img" focusable="false" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.826ex; min-width: 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transform="translate(889,0)"></use><use data-c="29" xlink:href="#MJX-121-TEX-N-29" transform="translate(1389,0)"></use></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(12)</mtext></mtd><mtd><mtable rowspacing=".5em" columnspacing="1em" displaystyle="true"><mtr><mtd><msub><mi>p</mi><mi>i</mi></msub><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mn>1</mn><msub><mi>γ</mi><mi>i</mi></msub></mfrac></mrow><mo stretchy="false">[</mo><msub><mi>r</mi><mn>0</mn></msub><mo>−</mo><mo stretchy="false">(</mo><msub><mi>z</mi><mi>i</mi></msub><mo>+</mo><msub><mi>r</mi><mn>0</mn></msub><mo>−</mo><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>∈</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>r</mi><mn>0</mn></msub><mo>,</mo><mi>A</mi><msub><mi>r</mi><mn>0</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msup><mi>A</mi><mi>i</mi></msup><msub><mi>r</mi><mn>0</mn></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></mtd></mtr><mtr><mtd><msub><mrow data-mjx-texclass="ORD"><mi data-mjx-variant="-tex-calligraphic" mathvariant="script">K</mi></mrow><mi>i</mi></msub><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mrow data-mjx-texclass="ORD"><mi data-mjx-variant="-tex-calligraphic" mathvariant="script">K</mi></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>p</mi><mi>i</mi></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>r</mi><mn>0</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msup><mi>A</mi><mi>i</mi></msup><msub><mi>r</mi><mn>0</mn></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><mi>◻</mi></mtd></mtr></mtable></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><strong><span>Proposition 2:</span></strong><span> </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="10.145ex" height="2.687ex" role="img" focusable="false" viewBox="0 -841.7 4484.1 1187.6" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.783ex;"><defs><path id="MJX-177-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-177-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 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46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msubsup"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-177-TEX-I-1D443"></use></g><g data-mml-node="mi" transform="translate(839.5,363) scale(0.707)"><use data-c="1D447" xlink:href="#MJX-177-TEX-I-1D447"></use></g><g data-mml-node="TeXAtom" transform="translate(675,-287.9) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-177-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(345,0)"><use data-c="2212" xlink:href="#MJX-177-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1123,0)"><use data-c="31" xlink:href="#MJX-177-TEX-N-31"></use></g></g></g><g data-mml-node="msub" transform="translate(1872.6,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-177-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(484,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-177-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(2928.4,0)"><use data-c="3D" xlink:href="#MJX-177-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(3984.1,0)"><use data-c="30" xlink:href="#MJX-177-TEX-N-30"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><msub><mi>r</mi><mi>i</mi></msub><mo>=</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">P_{i-1}^Tr_i=0</script></p><p><strong><span>Proof: </span></strong><span> 由前得知，</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="18.228ex" height="2.016ex" role="img" 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stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>x</mi><mn>0</mn><mi>T</mi></msubsup><mi>A</mi><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>y</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mrow data-mjx-texclass="ORD"><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><msubsup><mi>y</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><msubsup><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mi>A</mi><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>y</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><msup><mi>b</mi><mi>T</mi></msup><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>y</mi><mrow 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data-mml-node="mtext" transform="translate(3091.2,0)"><use data-c="A0" xlink:href="#MJX-179-TEX-N-A0"></use></g><g data-mml-node="msubsup" transform="translate(3341.2,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-179-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(484,363) scale(0.707)"><use data-c="1D447" xlink:href="#MJX-179-TEX-I-1D447"></use></g><g data-mml-node="mi" transform="translate(484,-284.4) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-179-TEX-I-1D456"></use></g></g><g data-mml-node="msub" transform="translate(4373,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-179-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(484,-150) scale(0.707)"><use data-c="1D457" xlink:href="#MJX-179-TEX-I-1D457"></use></g></g><g data-mml-node="mo" transform="translate(5476.1,0)"><use data-c="3D" xlink:href="#MJX-179-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(6531.9,0)"><use data-c="30" xlink:href="#MJX-179-TEX-N-30"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">∀</mi><mi>j</mi><mo>≠</mo><mi>i</mi><mo>,</mo><mtext>&nbsp;</mtext><msubsup><mi>r</mi><mi>i</mi><mi>T</mi></msubsup><msub><mi>r</mi><mi>j</mi></msub><mo>=</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">\forall j\ne i,\space r_i^Tr_j=0</script></p><p><strong><span>Proof: </span></strong><span> 由命题二得 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="19.914ex" height="2.375ex" role="img" focusable="false" viewBox="0 -841.7 8802.2 1049.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.471ex;"><defs><path id="MJX-180-TEX-N-2200" d="M0 673Q0 684 7 689T20 694Q32 694 38 680T82 567L126 451H430L473 566Q483 593 494 622T512 668T519 685Q524 694 538 694Q556 692 556 674Q556 670 426 329T293 -15Q288 -22 278 -22T263 -15Q260 -11 131 328T0 673ZM414 410Q414 411 278 411T142 410L278 55L414 410Z"></path><path id="MJX-180-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-180-TEX-N-2208" d="M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z"></path><path id="MJX-180-TEX-C-4B" d="M194 618Q193 618 182 613T156 601T131 594Q113 594 113 605Q113 623 144 644Q154 650 205 676T267 703Q277 705 279 705Q295 705 295 691Q295 569 250 397Q225 306 197 217T151 81T128 25Q120 8 94 -7T47 -22Q32 -22 32 -10L64 76Q95 163 133 295T185 530Q198 611 194 618ZM331 429Q331 383 364 290T449 117T542 36Q574 36 607 51T652 103Q660 124 677 133T709 143Q727 143 727 128Q727 119 723 111Q704 56 639 17T497 -22H493Q463 -22 425 16Q401 40 382 71Q335 138 296 243T256 399Q256 434 288 473Q342 540 471 622T670 705Q691 704 703 696Q732 678 732 644Q732 613 714 600T677 586Q671 586 667 587T660 592T657 604V619Q657 647 629 647Q623 647 620 646Q576 635 495 583T365 482Q331 448 331 429Z"></path><path id="MJX-180-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 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381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-180-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-180-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="2200" xlink:href="#MJX-180-TEX-N-2200"></use></g><g data-mml-node="mi" transform="translate(556,0)"><use data-c="1D465" xlink:href="#MJX-180-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1405.8,0)"><use data-c="2208" xlink:href="#MJX-180-TEX-N-2208"></use></g><g data-mml-node="msub" transform="translate(2350.6,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="4B" xlink:href="#MJX-180-TEX-C-4B"></use></g></g><g data-mml-node="TeXAtom" transform="translate(795,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-180-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(345,0)"><use data-c="2212" xlink:href="#MJX-180-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1123,0)"><use data-c="31" xlink:href="#MJX-180-TEX-N-31"></use></g></g></g><g data-mml-node="mo" transform="translate(4343.2,0)"><use data-c="2C" xlink:href="#MJX-180-TEX-N-2C"></use></g><g data-mml-node="mtext" transform="translate(4787.9,0)"><use data-c="A0" xlink:href="#MJX-180-TEX-N-A0"></use></g><g data-mml-node="msup" transform="translate(5037.9,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-180-TEX-I-1D465"></use></g><g data-mml-node="mi" transform="translate(605,363) scale(0.707)"><use data-c="1D447" xlink:href="#MJX-180-TEX-I-1D447"></use></g></g><g data-mml-node="msub" transform="translate(6190.7,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-180-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(484,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-180-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(7246.4,0)"><use data-c="3D" xlink:href="#MJX-180-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(8302.2,0)"><use data-c="30" xlink:href="#MJX-180-TEX-N-30"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">∀</mi><mi>x</mi><mo>∈</mo><msub><mrow data-mjx-texclass="ORD"><mi data-mjx-variant="-tex-calligraphic" mathvariant="script">K</mi></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mtext>&nbsp;</mtext><msup><mi>x</mi><mi>T</mi></msup><msub><mi>r</mi><mi>i</mi></msub><mo>=</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">\forall x\in\mathcal{K}_{i-1},\space x^Tr_i=0</script><span> ，故只需证明 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.99ex" height="1.952ex" role="img" focusable="false" viewBox="0 -705 3089.5 862.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-181-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-181-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path id="MJX-181-TEX-N-2208" d="M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z"></path><path id="MJX-181-TEX-C-4B" d="M194 618Q193 618 182 613T156 601T131 594Q113 594 113 605Q113 623 144 644Q154 650 205 676T267 703Q277 705 279 705Q295 705 295 691Q295 569 250 397Q225 306 197 217T151 81T128 25Q120 8 94 -7T47 -22Q32 -22 32 -10L64 76Q95 163 133 295T185 530Q198 611 194 618ZM331 429Q331 383 364 290T449 117T542 36Q574 36 607 51T652 103Q660 124 677 133T709 143Q727 143 727 128Q727 119 723 111Q704 56 639 17T497 -22H493Q463 -22 425 16Q401 40 382 71Q335 138 296 243T256 399Q256 434 288 473Q342 540 471 622T670 705Q691 704 703 696Q732 678 732 644Q732 613 714 600T677 586Q671 586 667 587T660 592T657 604V619Q657 647 629 647Q623 647 620 646Q576 635 495 583T365 482Q331 448 331 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-181-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(484,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-181-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(1055.7,0)"><use data-c="2208" xlink:href="#MJX-181-TEX-N-2208"></use></g><g data-mml-node="msub" transform="translate(2000.5,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="4B" xlink:href="#MJX-181-TEX-C-4B"></use></g></g><g data-mml-node="mi" transform="translate(795,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-181-TEX-I-1D456"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mi>i</mi></msub><mo>∈</mo><msub><mrow data-mjx-texclass="ORD"><mi data-mjx-variant="-tex-calligraphic" mathvariant="script">K</mi></mrow><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">r_i\in \mathcal{K}_i</script><span> 。使用归纳法，当 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.929ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 2178.6 748" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path id="MJX-182-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path id="MJX-182-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-182-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-182-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(622.8,0)"><use data-c="3D" xlink:href="#MJX-182-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1678.6,0)"><use data-c="31" xlink:href="#MJX-182-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">i=1</script><span> 时显然，若对 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.677ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 2067.4 748" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path id="MJX-183-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>∈</mo><msub><mrow data-mjx-texclass="ORD"><mi data-mjx-variant="-tex-calligraphic" mathvariant="script">K</mi></mrow><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">Ap_{i-1}\in\mathcal{K}_i</script><span>，则 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="13.275ex" height="1.952ex" role="img" focusable="false" viewBox="0 -705 5867.5 862.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-187-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-187-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path id="MJX-187-TEX-N-2208" d="M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z"></path><path id="MJX-187-TEX-C-4B" d="M194 618Q193 618 182 613T156 601T131 594Q113 594 113 605Q113 623 144 644Q154 650 205 676T267 703Q277 705 279 705Q295 705 295 691Q295 569 250 397Q225 306 197 217T151 81T128 25Q120 8 94 -7T47 -22Q32 -22 32 -10L64 76Q95 163 133 295T185 530Q198 611 194 618ZM331 429Q331 383 364 290T449 117T542 36Q574 36 607 51T652 103Q660 124 677 133T709 143Q727 143 727 128Q727 119 723 111Q704 56 639 17T497 -22H493Q463 -22 425 16Q401 40 382 71Q335 138 296 243T256 399Q256 434 288 473Q342 540 471 622T670 705Q691 704 703 696Q732 678 732 644Q732 613 714 600T677 586Q671 586 667 587T660 592T657 604V619Q657 647 629 647Q623 647 620 646Q576 635 495 583T365 482Q331 448 331 429Z"></path><path id="MJX-187-TEX-I-25FB" d="M71 0Q59 4 55 16V346L56 676Q64 686 70 689H709Q719 681 722 674V15Q719 10 709 1L390 0H71ZM682 40V649H95V40H682Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-187-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(484,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-187-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(1055.7,0)"><use data-c="2208" xlink:href="#MJX-187-TEX-N-2208"></use></g><g data-mml-node="msub" transform="translate(2000.5,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="4B" xlink:href="#MJX-187-TEX-C-4B"></use></g></g><g data-mml-node="mi" transform="translate(795,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-187-TEX-I-1D456"></use></g></g><g data-mml-node="mstyle" transform="translate(3089.5,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mi" transform="translate(5089.5,0)"><use data-c="25FB" xlink:href="#MJX-187-TEX-I-25FB"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mi>i</mi></msub><mo>∈</mo><msub><mrow data-mjx-texclass="ORD"><mi data-mjx-variant="-tex-calligraphic" mathvariant="script">K</mi></mrow><mi>i</mi></msub><mstyle scriptlevel="0"><mspace width="2em"></mspace></mstyle><mi>◻</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">r_i\in\mathcal{K}_i\qquad\square</script></p><p><span>由命题三可知，</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="16.05ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 7094 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-188-TEX-C-4B" d="M194 618Q193 618 182 613T156 601T131 594Q113 594 113 605Q113 623 144 644Q154 650 205 676T267 703Q277 705 279 705Q295 705 295 691Q295 569 250 397Q225 306 197 217T151 81T128 25Q120 8 94 -7T47 -22Q32 -22 32 -10L64 76Q95 163 133 295T185 530Q198 611 194 618ZM331 429Q331 383 364 290T449 117T542 36Q574 36 607 51T652 103Q660 124 677 133T709 143Q727 143 727 128Q727 119 723 111Q704 56 639 17T497 -22H493Q463 -22 425 16Q401 40 382 71Q335 138 296 243T256 399Q256 434 288 473Q342 540 471 622T670 705Q691 704 703 696Q732 678 732 644Q732 613 714 600T677 586Q671 586 667 587T660 592T657 604V619Q657 647 629 647Q623 647 620 646Q576 635 495 583T365 482Q331 448 331 429Z"></path><path id="MJX-188-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path id="MJX-188-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-188-TEX-N-27E8" d="M333 -232Q332 -239 327 -244T313 -250Q303 -250 296 -240Q293 -233 202 6T110 250T201 494T296 740Q299 745 306 749L309 750Q312 750 313 750Q331 750 333 732Q333 727 243 489Q152 252 152 250T243 11Q333 -227 333 -232Z"></path><path id="MJX-188-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-188-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-188-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-188-TEX-N-22EF" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250ZM525 250Q525 274 542 292T585 310Q609 310 627 294T646 251Q646 226 629 208T586 190T543 207T525 250ZM972 250Q972 274 989 292T1032 310Q1056 310 1074 294T1093 251Q1093 226 1076 208T1033 190T990 207T972 250Z"></path><path id="MJX-188-TEX-N-27E9" d="M55 732Q56 739 61 744T75 750Q85 750 92 740Q95 733 186 494T278 250T187 6T92 -240Q85 -250 75 -250Q67 -250 62 -245T55 -232Q55 -227 145 11Q236 248 236 250T145 489Q55 727 55 732Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="4B" xlink:href="#MJX-188-TEX-C-4B"></use></g></g><g data-mml-node="mi" transform="translate(795,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-188-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(1366.7,0)"><use data-c="3D" xlink:href="#MJX-188-TEX-N-3D"></use></g><g data-mml-node="mrow" transform="translate(2422.5,0)"><g data-mml-node="mo"><use data-c="27E8" xlink:href="#MJX-188-TEX-N-27E8"></use></g><g data-mml-node="msub" transform="translate(389,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-188-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-188-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(1276.6,0)"><use data-c="2C" xlink:href="#MJX-188-TEX-N-2C"></use></g><g data-mml-node="mo" transform="translate(1721.2,0)"><use data-c="22EF" xlink:href="#MJX-188-TEX-N-22EF"></use></g><g data-mml-node="mo" transform="translate(3059.9,0)"><use data-c="2C" xlink:href="#MJX-188-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(3504.6,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-188-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(484,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-188-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(4282.5,0)"><use data-c="27E9" xlink:href="#MJX-188-TEX-N-27E9"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi data-mjx-variant="-tex-calligraphic" mathvariant="script">K</mi></mrow><mi>i</mi></msub><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>r</mi><mn>0</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>r</mi><mi>i</mi></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\mathcal{K}_i=\left<r_0,\cdots,r_i\right></script><span>, 且 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="8.809ex" height="1.439ex" role="img" focusable="false" viewBox="0 -442 3893.5 636" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-189-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-189-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-189-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-189-TEX-N-22EF" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250ZM525 250Q525 274 542 292T585 310Q609 310 627 294T646 251Q646 226 629 208T586 190T543 207T525 250ZM972 250Q972 274 989 292T1032 310Q1056 310 1074 294T1093 251Q1093 226 1076 208T1033 190T990 207T972 250Z"></path><path id="MJX-189-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-189-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-189-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(887.6,0)"><use data-c="2C" xlink:href="#MJX-189-TEX-N-2C"></use></g><g data-mml-node="mo" transform="translate(1332.2,0)"><use data-c="22EF" xlink:href="#MJX-189-TEX-N-22EF"></use></g><g data-mml-node="mo" transform="translate(2670.9,0)"><use data-c="2C" xlink:href="#MJX-189-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(3115.6,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-189-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(484,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-189-TEX-I-1D456"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>0</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>r</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">r_0,\cdots,r_i</script><span> 构成一组正交基。</span></p><p><strong><span>Proposition 4:</span></strong><span>   </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="13.092ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 5786.8 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-190-TEX-I-1D45D" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 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data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">p_i\in\left<r_i,p_{i-1}\right></script></p><p><strong><span>Proof: </span></strong><span> 首先，</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.792ex" height="1.357ex" role="img" focusable="false" viewBox="0 -442 792 599.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-191-TEX-I-1D467" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 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rowspacing="3pt"><mtr><mtd><msub><mi>γ</mi><mi>i</mi></msub><msub><mi>p</mi><mi>i</mi></msub></mtd><mtd><mi></mi><mo>=</mo><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mi>A</mi><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mrow data-mjx-texclass="ORD"><mfrac><msub><mi>β</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mfrac></mrow><mo stretchy="false">(</mo><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>γ</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>−</mo><msub><mi>β</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mi>A</mi><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mn>1</mn><mo>+</mo><mrow data-mjx-texclass="ORD"><mfrac><msub><mi>β</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><msub><mi>r</mi><mi>i</mi></msub><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><msub><mi>β</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mfrac></mrow><msub><mi>γ</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mstyle scriptlevel="0"><mspace width="2em"></mspace></mstyle><mstyle scriptlevel="0"><mspace width="2em"></mspace></mstyle><mstyle scriptlevel="0"><mspace width="2em"></mspace></mstyle><mi>◻</mi></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>最终实现</span></p><pre class="md-fences md-end-block ty-contain-cm modeLoaded" spellcheck="false" lang="python" style="break-inside: unset;"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 12px; left: 4px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><span><span>​</span>x</span></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">import</span> <span class="cm-variable">numpy</span> <span class="cm-keyword">as</span> <span class="cm-variable">np</span></span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">CG</span>(<span class="cm-variable">A</span>,<span class="cm-variable">b</span>,<span class="cm-variable">x</span>,<span class="cm-variable">n</span>):</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">r</span> = <span class="cm-variable">b</span> <span class="cm-operator">-</span> <span class="cm-variable">A</span> <span class="cm-operator">@</span> <span class="cm-variable">x</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">i</span> <span class="cm-keyword">in</span> <span class="cm-builtin">range</span>(<span class="cm-variable">n</span>):</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">s</span> = <span class="cm-variable">r</span> <span class="cm-operator">@</span> <span class="cm-variable">r</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">if</span> <span class="cm-variable">i</span>==<span class="cm-number">0</span>:</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">p</span> = <span class="cm-variable">r</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">else</span>:</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">p</span> = <span class="cm-variable">r</span> <span class="cm-operator">+</span> <span class="cm-variable">s</span> <span class="cm-operator">/</span> <span class="cm-variable">_s</span> <span class="cm-operator">*</span> <span class="cm-variable">_p</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">_s</span>, <span class="cm-variable">_p</span> = <span class="cm-variable">s</span>, <span class="cm-variable">p</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">q</span> = <span class="cm-variable">A</span> <span class="cm-operator">@</span> <span class="cm-variable">p</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">a</span> = <span class="cm-variable">s</span> <span class="cm-operator">/</span> (<span class="cm-variable">q</span> <span class="cm-operator">@</span> <span class="cm-variable">p</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">x</span> += <span class="cm-variable">a</span> <span class="cm-operator">*</span> <span class="cm-variable">p</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">r</span> -= <span class="cm-variable">a</span> <span class="cm-operator">*</span> <span class="cm-variable">q</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-builtin">print</span>(<span class="cm-builtin">max</span>(<span class="cm-builtin">abs</span>(<span class="cm-variable">r</span>)))</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">return</span> <span class="cm-variable">x</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 504px;"></div><div class="CodeMirror-gutters" style="display: none; height: 504px;"></div></div></div></pre><p>&nbsp;</p><h2 id='bicg'><span>BiCG</span></h2><p><span>Let </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="22.86ex" height="2.715ex" role="img" focusable="false" viewBox="0 -850 10104 1200" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.792ex;"><defs><path id="MJX-196-TEX-C-4B" d="M194 618Q193 618 182 613T156 601T131 594Q113 594 113 605Q113 623 144 644Q154 650 205 676T267 703Q277 705 279 705Q295 705 295 691Q295 569 250 397Q225 306 197 217T151 81T128 25Q120 8 94 -7T47 -22Q32 -22 32 -10L64 76Q95 163 133 295T185 530Q198 611 194 618ZM331 429Q331 383 364 290T449 117T542 36Q574 36 607 51T652 103Q660 124 677 133T709 143Q727 143 727 128Q727 119 723 111Q704 56 639 17T497 -22H493Q463 -22 425 16Q401 40 382 71Q335 138 296 243T256 399Q256 434 288 473Q342 540 471 622T670 705Q691 704 703 696Q732 678 732 644Q732 613 714 600T677 586Q671 586 667 587T660 592T657 604V619Q657 647 629 647Q623 647 620 646Q576 635 495 583T365 482Q331 448 331 429Z"></path><path id="MJX-196-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-196-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-196-TEX-SO-27E8" d="M373 850Q392 850 394 832Q394 825 267 538L139 250L267 -38Q394 -325 394 -332Q392 -350 375 -350Q361 -350 356 -338Q354 -331 289 -186T161 103T97 250T160 397T289 685T356 838Q362 850 373 850Z"></path><path id="MJX-196-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-196-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-196-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-196-TEX-N-22EF" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250ZM525 250Q525 274 542 292T585 310Q609 310 627 294T646 251Q646 226 629 208T586 190T543 207T525 250ZM972 250Q972 274 989 292T1032 310Q1056 310 1074 294T1093 251Q1093 226 1076 208T1033 190T990 207T972 250Z"></path><path id="MJX-196-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-196-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-196-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-196-TEX-SO-27E9" d="M77 832Q77 837 82 843T98 850Q110 849 115 838Q117 831 182 686T310 397T374 250T311 103T182 -185T115 -338Q110 -350 96 -350Q79 -350 77 -332Q77 -325 204 -38L332 250L204 538Q77 825 77 832Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="4B" xlink:href="#MJX-196-TEX-C-4B"></use></g></g><g data-mml-node="mi" transform="translate(795,-150) scale(0.707)"><use data-c="1D45A" xlink:href="#MJX-196-TEX-I-1D45A"></use></g></g><g data-mml-node="mo" transform="translate(1743.6,0)"><use data-c="3D" xlink:href="#MJX-196-TEX-N-3D"></use></g><g data-mml-node="mrow" transform="translate(2799.4,0)"><g data-mml-node="mo"><use data-c="27E8" xlink:href="#MJX-196-TEX-SO-27E8"></use></g><g data-mml-node="msub" transform="translate(472,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-196-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-196-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(1359.6,0)"><use data-c="2C" xlink:href="#MJX-196-TEX-N-2C"></use></g><g data-mml-node="mo" transform="translate(1804.2,0)"><use data-c="22EF" xlink:href="#MJX-196-TEX-N-22EF"></use></g><g data-mml-node="mo" transform="translate(3142.9,0)"><use data-c="2C" xlink:href="#MJX-196-TEX-N-2C"></use></g><g data-mml-node="msup" transform="translate(3587.6,0)"><g data-mml-node="mi"><use data-c="1D434" xlink:href="#MJX-196-TEX-I-1D434"></use></g><g data-mml-node="TeXAtom" transform="translate(783,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D45A" xlink:href="#MJX-196-TEX-I-1D45A"></use></g><g data-mml-node="mo" transform="translate(878,0)"><use data-c="2212" xlink:href="#MJX-196-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1656,0)"><use data-c="31" xlink:href="#MJX-196-TEX-N-31"></use></g></g></g><g data-mml-node="msub" transform="translate(5945.1,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-196-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-196-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(6832.6,0)"><use data-c="27E9" xlink:href="#MJX-196-TEX-SO-27E9"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi data-mjx-variant="-tex-calligraphic" mathvariant="script">K</mi></mrow><mi>m</mi></msub><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>r</mi><mn>0</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msup><mi>A</mi><mrow data-mjx-texclass="ORD"><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mi>r</mi><mn>0</mn></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\mathcal{K}_m=\left<r_0,\cdots,A^{m-1}r_0\right></script><span>,  </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="26.712ex" height="4.07ex" role="img" focusable="false" viewBox="0 -1149.5 11806.8 1799" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -1.469ex;"><defs><path id="MJX-197-TEX-C-4C" d="M62 -22T47 -22T32 -11Q32 -1 56 24T83 55Q113 96 138 172T180 320T234 473T323 609Q364 649 419 677T531 705Q559 705 578 696T604 671T615 645T618 623V611Q618 582 615 571T598 548Q581 531 558 520T518 509Q503 509 503 520Q503 523 505 536T507 560Q507 590 494 610T452 630Q423 630 410 617Q367 578 333 492T271 301T233 170Q211 123 204 112L198 103L224 102Q281 102 369 79T509 52H523Q535 64 544 87T579 128Q616 152 641 152Q656 152 656 142Q656 101 588 40T433 -22Q381 -22 289 1T156 28L141 29L131 20Q111 0 87 -11Z"></path><path id="MJX-197-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-197-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-197-TEX-LO-27E8" d="M112 244V258L473 1130Q482 1150 498 1150Q511 1150 517 1142T523 1125V1118L344 685Q304 587 257 473T187 305L165 251L344 -184L523 -616V-623Q524 -634 517 -641T499 -649Q484 -649 473 -629L112 244Z"></path><path id="MJX-197-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-197-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-197-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-197-TEX-N-22EF" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250ZM525 250Q525 274 542 292T585 310Q609 310 627 294T646 251Q646 226 629 208T586 190T543 207T525 250ZM972 250Q972 274 989 292T1032 310Q1056 310 1074 294T1093 251Q1093 226 1076 208T1033 190T990 207T972 250Z"></path><path id="MJX-197-TEX-SO-28" d="M152 251Q152 646 388 850H416Q422 844 422 841Q422 837 403 816T357 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630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path id="MJX-197-TEX-SO-29" d="M305 251Q305 -145 69 -349H56Q43 -349 39 -347T35 -338Q37 -333 60 -307T108 -239T160 -136T204 27T221 250T204 473T160 636T108 740T60 807T35 839Q35 850 50 850H56H69Q197 743 256 566Q305 425 305 251Z"></path><path id="MJX-197-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-197-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-197-TEX-LO-27E9" d="M112 -649Q103 -649 95 -642T87 -623V-616L266 -184L445 251Q445 252 356 466T178 898T86 1123Q85 1134 93 1142T110 1150Q126 1150 133 1137Q134 1136 317 695L498 258V244L317 -194Q134 -635 133 -636Q126 -649 112 -649Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="4C" xlink:href="#MJX-197-TEX-C-4C"></use></g></g><g data-mml-node="mi" transform="translate(723,-150) scale(0.707)"><use data-c="1D45A" xlink:href="#MJX-197-TEX-I-1D45A"></use></g></g><g data-mml-node="mo" transform="translate(1671.6,0)"><use data-c="3D" xlink:href="#MJX-197-TEX-N-3D"></use></g><g data-mml-node="mrow" transform="translate(2727.4,0)"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="27E8" xlink:href="#MJX-197-TEX-LO-27E8"></use></g><g data-mml-node="msub" transform="translate(611,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-197-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-197-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(1498.6,0)"><use data-c="2C" xlink:href="#MJX-197-TEX-N-2C"></use></g><g data-mml-node="mo" transform="translate(1943.2,0)"><use data-c="22EF" xlink:href="#MJX-197-TEX-N-22EF"></use></g><g data-mml-node="mo" transform="translate(3281.9,0)"><use data-c="2C" xlink:href="#MJX-197-TEX-N-2C"></use></g><g data-mml-node="msup" transform="translate(3726.6,0)"><g data-mml-node="mrow"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="28" xlink:href="#MJX-197-TEX-SO-28"></use></g><g data-mml-node="msup" transform="translate(458,0)"><g data-mml-node="mi"><use data-c="1D434" xlink:href="#MJX-197-TEX-I-1D434"></use></g><g data-mml-node="TeXAtom" transform="translate(783,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D447" xlink:href="#MJX-197-TEX-I-1D447"></use></g></g></g><g data-mml-node="mo" transform="translate(1788.8,0) translate(0 -0.5)"><use data-c="29" xlink:href="#MJX-197-TEX-SO-29"></use></g></g><g data-mml-node="TeXAtom" transform="translate(2279.8,576.6) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D45A" xlink:href="#MJX-197-TEX-I-1D45A"></use></g><g data-mml-node="mo" transform="translate(878,0)"><use data-c="2212" xlink:href="#MJX-197-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1656,0)"><use data-c="31" xlink:href="#MJX-197-TEX-N-31"></use></g></g></g><g data-mml-node="msub" transform="translate(7580.9,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-197-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-197-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(8468.4,0) translate(0 -0.5)"><use data-c="27E9" xlink:href="#MJX-197-TEX-LO-27E9"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi data-mjx-variant="-tex-calligraphic" mathvariant="script">L</mi></mrow><mi>m</mi></msub><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>r</mi><mn>0</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>A</mi><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mi>r</mi><mn>0</mn></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\mathcal{L}_m=\left<r_0,\cdots,\left(A^{T}\right)^{m-1}r_0\right></script><span>, let </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="18.006ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 7958.8 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-198-TEX-I-1D449" d="M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z"></path><path id="MJX-198-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 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transform="translate(518,-150) scale(0.707)"><use data-c="1D45A" xlink:href="#MJX-198-TEX-I-1D45A"></use></g></g><g data-mml-node="mo" transform="translate(7458.8,0)"><use data-c="7D" xlink:href="#MJX-198-TEX-N-7D"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>m</mi></msub><mo>=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>v</mi><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>v</mi><mi>m</mi></msub><mo fence="false" stretchy="false">}</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">V_m=\{v_1,\cdots,v_m\}</script><span> be a set of basis of </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.316ex" height="1.952ex" role="img" focusable="false" viewBox="0 -705 1465.8 862.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-199-TEX-C-4B" d="M194 618Q193 618 182 613T156 601T131 594Q113 594 113 605Q113 623 144 644Q154 650 205 676T267 703Q277 705 279 705Q295 705 295 691Q295 569 250 397Q225 306 197 217T151 81T128 25Q120 8 94 -7T47 -22Q32 -22 32 -10L64 76Q95 163 133 295T185 530Q198 611 194 618ZM331 429Q331 383 364 290T449 117T542 36Q574 36 607 51T652 103Q660 124 677 133T709 143Q727 143 727 128Q727 119 723 111Q704 56 639 17T497 -22H493Q463 -22 425 16Q401 40 382 71Q335 138 296 243T256 399Q256 434 288 473Q342 540 471 622T670 705Q691 704 703 696Q732 678 732 644Q732 613 714 600T677 586Q671 586 667 587T660 592T657 604V619Q657 647 629 647Q623 647 620 646Q576 635 495 583T365 482Q331 448 331 429Z"></path><path id="MJX-199-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="4B" xlink:href="#MJX-199-TEX-C-4B"></use></g></g><g data-mml-node="mi" transform="translate(795,-150) scale(0.707)"><use data-c="1D45A" 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transform="translate(6861.9,0)"><g data-mml-node="mi"><use data-c="1D464" xlink:href="#MJX-200-TEX-I-1D464"></use></g><g data-mml-node="mi" transform="translate(749,-150) scale(0.707)"><use data-c="1D45A" xlink:href="#MJX-200-TEX-I-1D45A"></use></g></g><g data-mml-node="mo" transform="translate(8281.8,0)"><use data-c="7D" xlink:href="#MJX-200-TEX-N-7D"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>W</mi><mi>m</mi></msub><mo>=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>w</mi><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>w</mi><mi>m</mi></msub><mo fence="false" stretchy="false">}</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">W_m=\{ w_1,\cdots,w_m \} </script><span> be a set of basis of </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.153ex" height="1.952ex" role="img" focusable="false" viewBox="0 -705 1393.8 862.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-211-TEX-C-4C" d="M62 -22T47 -22T32 -11Q32 -1 56 24T83 55Q113 96 138 172T180 320T234 473T323 609Q364 649 419 677T531 705Q559 705 578 696T604 671T615 645T618 623V611Q618 582 615 571T598 548Q581 531 558 520T518 509Q503 509 503 520Q503 523 505 536T507 560Q507 590 494 610T452 630Q423 630 410 617Q367 578 333 492T271 301T233 170Q211 123 204 112L198 103L224 102Q281 102 369 79T509 52H523Q535 64 544 87T579 128Q616 152 641 152Q656 152 656 142Q656 101 588 40T433 -22Q381 -22 289 1T156 28L141 29L131 20Q111 0 87 -11Z"></path><path id="MJX-211-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="4C" xlink:href="#MJX-211-TEX-C-4C"></use></g></g><g data-mml-node="mi" transform="translate(723,-150) scale(0.707)"><use data-c="1D45A" xlink:href="#MJX-211-TEX-I-1D45A"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi data-mjx-variant="-tex-calligraphic" mathvariant="script">L</mi></mrow><mi>m</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\mathcal{L}_m</script><span>.  We further require that </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.911ex" height="1.902ex" role="img" focusable="false" viewBox="0 -683 1286.8 840.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-202-TEX-I-1D449" d="M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z"></path><path id="MJX-202-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D449" xlink:href="#MJX-202-TEX-I-1D449"></use></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><use data-c="1D45A" xlink:href="#MJX-202-TEX-I-1D45A"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>m</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">V_m</script><span> and </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.728ex" height="1.902ex" role="img" focusable="false" viewBox="0 -683 1647.8 840.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-203-TEX-I-1D44A" d="M436 683Q450 683 486 682T553 680Q604 680 638 681T677 682Q695 682 695 674Q695 670 692 659Q687 641 683 639T661 637Q636 636 621 632T600 624T597 615Q597 603 613 377T629 138L631 141Q633 144 637 151T649 170T666 200T690 241T720 295T759 362Q863 546 877 572T892 604Q892 619 873 628T831 637Q817 637 817 647Q817 650 819 660Q823 676 825 679T839 682Q842 682 856 682T895 682T949 681Q1015 681 1034 683Q1048 683 1048 672Q1048 666 1045 655T1038 640T1028 637Q1006 637 988 631T958 617T939 600T927 584L923 578L754 282Q586 -14 585 -15Q579 -22 561 -22Q546 -22 542 -17Q539 -14 523 229T506 480L494 462Q472 425 366 239Q222 -13 220 -15T215 -19Q210 -22 197 -22Q178 -22 176 -15Q176 -12 154 304T131 622Q129 631 121 633T82 637H58Q51 644 51 648Q52 671 64 683H76Q118 680 176 680Q301 680 313 683H323Q329 677 329 674T327 656Q322 641 318 637H297Q236 634 232 620Q262 160 266 136L501 550L499 587Q496 629 489 632Q483 636 447 637Q428 637 422 639T416 648Q416 650 418 660Q419 664 420 669T421 676T424 680T428 682T436 683Z"></path><path id="MJX-203-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D44A" xlink:href="#MJX-203-TEX-I-1D44A"></use></g><g data-mml-node="mi" transform="translate(977,-150) scale(0.707)"><use data-c="1D45A" xlink:href="#MJX-203-TEX-I-1D45A"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>W</mi><mi>m</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">W_m</script><span> satisfies </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="12.326ex" height="2.481ex" role="img" focusable="false" viewBox="0 -841.7 5448.2 1096.5" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.576ex;"><defs><path id="MJX-204-TEX-I-1D44A" d="M436 683Q450 683 486 682T553 680Q604 680 638 681T677 682Q695 682 695 674Q695 670 692 659Q687 641 683 639T661 637Q636 636 621 632T600 624T597 615Q597 603 613 377T629 138L631 141Q633 144 637 151T649 170T666 200T690 241T720 295T759 362Q863 546 877 572T892 604Q892 619 873 628T831 637Q817 637 817 647Q817 650 819 660Q823 676 825 679T839 682Q842 682 856 682T895 682T949 681Q1015 681 1034 683Q1048 683 1048 672Q1048 666 1045 655T1038 640T1028 637Q1006 637 988 631T958 617T939 600T927 584L923 578L754 282Q586 -14 585 -15Q579 -22 561 -22Q546 -22 542 -17Q539 -14 523 229T506 480L494 462Q472 425 366 239Q222 -13 220 -15T215 -19Q210 -22 197 -22Q178 -22 176 -15Q176 -12 154 304T131 622Q129 631 121 633T82 637H58Q51 644 51 648Q52 671 64 683H76Q118 680 176 680Q301 680 313 683H323Q329 677 329 674T327 656Q322 641 318 637H297Q236 634 232 620Q262 160 266 136L501 550L499 587Q496 629 489 632Q483 636 447 637Q428 637 422 639T416 648Q416 650 418 660Q419 664 420 669T421 676T424 680T428 682T436 683Z"></path><path id="MJX-204-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path id="MJX-204-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-204-TEX-I-1D449" d="M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z"></path><path id="MJX-204-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-204-TEX-I-1D43C" d="M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msubsup"><g data-mml-node="mi"><use data-c="1D44A" xlink:href="#MJX-204-TEX-I-1D44A"></use></g><g data-mml-node="mi" transform="translate(1136.2,363) scale(0.707)"><use data-c="1D447" xlink:href="#MJX-204-TEX-I-1D447"></use></g><g data-mml-node="mi" transform="translate(977,-247) scale(0.707)"><use data-c="1D45A" xlink:href="#MJX-204-TEX-I-1D45A"></use></g></g><g data-mml-node="msub" transform="translate(1684,0)"><g data-mml-node="mi"><use data-c="1D449" xlink:href="#MJX-204-TEX-I-1D449"></use></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><use data-c="1D45A" xlink:href="#MJX-204-TEX-I-1D45A"></use></g></g><g data-mml-node="mo" transform="translate(3248.6,0)"><use data-c="3D" xlink:href="#MJX-204-TEX-N-3D"></use></g><g data-mml-node="msub" transform="translate(4304.4,0)"><g data-mml-node="mi"><use data-c="1D43C" xlink:href="#MJX-204-TEX-I-1D43C"></use></g><g data-mml-node="mi" transform="translate(473,-150) scale(0.707)"><use data-c="1D45A" xlink:href="#MJX-204-TEX-I-1D45A"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>W</mi><mi>m</mi><mi>T</mi></msubsup><msub><mi>V</mi><mi>m</mi></msub><mo>=</mo><msub><mi>I</mi><mi>m</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">W_m^TV_m=I_m</script><span> .  Then </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="8.418ex" height="2.481ex" role="img" focusable="false" viewBox="0 -841.7 3720.8 1096.5" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.576ex;"><defs><path id="MJX-205-TEX-I-1D44A" d="M436 683Q450 683 486 682T553 680Q604 680 638 681T677 682Q695 682 695 674Q695 670 692 659Q687 641 683 639T661 637Q636 636 621 632T600 624T597 615Q597 603 613 377T629 138L631 141Q633 144 637 151T649 170T666 200T690 241T720 295T759 362Q863 546 877 572T892 604Q892 619 873 628T831 637Q817 637 817 647Q817 650 819 660Q823 676 825 679T839 682Q842 682 856 682T895 682T949 681Q1015 681 1034 683Q1048 683 1048 672Q1048 666 1045 655T1038 640T1028 637Q1006 637 988 631T958 617T939 600T927 584L923 578L754 282Q586 -14 585 -15Q579 -22 561 -22Q546 -22 542 -17Q539 -14 523 229T506 480L494 462Q472 425 366 239Q222 -13 220 -15T215 -19Q210 -22 197 -22Q178 -22 176 -15Q176 -12 154 304T131 622Q129 631 121 633T82 637H58Q51 644 51 648Q52 671 64 683H76Q118 680 176 680Q301 680 313 683H323Q329 677 329 674T327 656Q322 641 318 637H297Q236 634 232 620Q262 160 266 136L501 550L499 587Q496 629 489 632Q483 636 447 637Q428 637 422 639T416 648Q416 650 418 660Q419 664 420 669T421 676T424 680T428 682T436 683Z"></path><path id="MJX-205-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path id="MJX-205-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-205-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-205-TEX-I-1D449" d="M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 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xlink:href="#MJX-131-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(521,0)"><use data-c="2212" xlink:href="#MJX-131-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1299,0)"><use data-c="31" xlink:href="#MJX-131-TEX-N-31"></use></g></g></g><rect width="2052.1" height="60" x="120" y="220"></rect></g></g></g></g></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1778 -2689 1 4878"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-mjx-eqn:20"><g data-mml-node="mtext"><use data-c="28" xlink:href="#MJX-131-TEX-N-28"></use><use data-c="32" xlink:href="#MJX-131-TEX-N-32" transform="translate(389,0)"></use><use data-c="30" xlink:href="#MJX-131-TEX-N-30" transform="translate(889,0)"></use><use data-c="29" xlink:href="#MJX-131-TEX-N-29" transform="translate(1389,0)"></use></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(20)</mtext></mtd><mtd><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><msub><mi>η</mi><mi>k</mi></msub></mtd><mtd><mi></mi><mo>=</mo><msub><mi>α</mi><mi>k</mi></msub><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow><msub><mi>β</mi><mi>k</mi></msub><msub><mi>δ</mi><mi>k</mi></msub></mrow><msub><mi>η</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mfrac></mrow><mo>,</mo><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><msub><mi>η</mi><mn>0</mn></msub><mo>=</mo><msub><mi>α</mi><mn>0</mn></msub></mtd></mtr><mtr><mtd><msub><mi>λ</mi><mi>k</mi></msub></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><msub><mi>β</mi><mi>k</mi></msub><msub><mi>η</mi><mrow 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xlink:href="#MJX-209-TEX-I-1D45A"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mi>m</mi></msub><mo>∈</mo><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><msub><mrow data-mjx-texclass="ORD"><mi data-mjx-variant="-tex-calligraphic" mathvariant="script">K</mi></mrow><mi>m</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">x_m\in x_0+\mathcal{K}_m</script><span>,  such that </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="13.95ex" height="1.977ex" role="img" focusable="false" viewBox="0 -716 6165.7 873.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-210-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-210-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-210-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-210-TEX-I-1D44F" d="M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z"></path><path id="MJX-210-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-210-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-210-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-210-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(484,-150) scale(0.707)"><use data-c="1D45A" xlink:href="#MJX-210-TEX-I-1D45A"></use></g></g><g data-mml-node="mo" transform="translate(1432.6,0)"><use data-c="3D" xlink:href="#MJX-210-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(2488.4,0)"><use data-c="1D44F" xlink:href="#MJX-210-TEX-I-1D44F"></use></g><g data-mml-node="mo" transform="translate(3139.6,0)"><use data-c="2212" xlink:href="#MJX-210-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(4139.8,0)"><use data-c="1D434" xlink:href="#MJX-210-TEX-I-1D434"></use></g><g data-mml-node="msub" transform="translate(4889.8,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-210-TEX-I-1D465"></use></g><g data-mml-node="mi" transform="translate(605,-150) scale(0.707)"><use data-c="1D45A" xlink:href="#MJX-210-TEX-I-1D45A"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mi>m</mi></msub><mo>=</mo><mi>b</mi><mo>−</mo><mi>A</mi><msub><mi>x</mi><mi>m</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">r_m=b-Ax_m</script><span> is orthogonal to </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.153ex" height="1.952ex" role="img" focusable="false" viewBox="0 -705 1393.8 862.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-211-TEX-C-4C" d="M62 -22T47 -22T32 -11Q32 -1 56 24T83 55Q113 96 138 172T180 320T234 473T323 609Q364 649 419 677T531 705Q559 705 578 696T604 671T615 645T618 623V611Q618 582 615 571T598 548Q581 531 558 520T518 509Q503 509 503 520Q503 523 505 536T507 560Q507 590 494 610T452 630Q423 630 410 617Q367 578 333 492T271 301T233 170Q211 123 204 112L198 103L224 102Q281 102 369 79T509 52H523Q535 64 544 87T579 128Q616 152 641 152Q656 152 656 142Q656 101 588 40T433 -22Q381 -22 289 1T156 28L141 29L131 20Q111 0 87 -11Z"></path><path id="MJX-211-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="4C" xlink:href="#MJX-211-TEX-C-4C"></use></g></g><g data-mml-node="mi" transform="translate(723,-150) scale(0.707)"><use data-c="1D45A" xlink:href="#MJX-211-TEX-I-1D45A"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi data-mjx-variant="-tex-calligraphic" mathvariant="script">L</mi></mrow><mi>m</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\mathcal{L}_m</script><span> . Let </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="16.564ex" height="2.009ex" role="img" focusable="false" viewBox="0 -683 7321.1 888" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-212-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-212-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-212-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-212-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-212-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-212-TEX-I-1D449" d="M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z"></path><path id="MJX-212-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-212-TEX-I-1D465"></use></g><g data-mml-node="mi" transform="translate(605,-150) scale(0.707)"><use data-c="1D45A" xlink:href="#MJX-212-TEX-I-1D45A"></use></g></g><g data-mml-node="mo" transform="translate(1553.6,0)"><use data-c="3D" xlink:href="#MJX-212-TEX-N-3D"></use></g><g data-mml-node="msub" transform="translate(2609.4,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-212-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-212-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(3840.2,0)"><use data-c="2B" xlink:href="#MJX-212-TEX-N-2B"></use></g><g data-mml-node="msub" transform="translate(4840.4,0)"><g data-mml-node="mi"><use data-c="1D449" xlink:href="#MJX-212-TEX-I-1D449"></use></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><use data-c="1D45A" xlink:href="#MJX-212-TEX-I-1D45A"></use></g></g><g data-mml-node="msub" transform="translate(6127.2,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-212-TEX-I-1D466"></use></g><g data-mml-node="mi" transform="translate(523,-150) scale(0.707)"><use data-c="1D45A" xlink:href="#MJX-212-TEX-I-1D45A"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mi>m</mi></msub><mo>=</mo><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><msub><mi>V</mi><mi>m</mi></msub><msub><mi>y</mi><mi>m</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">x_m=x_0+V_my_m</script><span>, then </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="31.241ex" height="2.481ex" role="img" focusable="false" viewBox="0 -841.7 13808.6 1096.5" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.576ex;"><defs><path id="MJX-213-TEX-I-1D44A" d="M436 683Q450 683 486 682T553 680Q604 680 638 681T677 682Q695 682 695 674Q695 670 692 659Q687 641 683 639T661 637Q636 636 621 632T600 624T597 615Q597 603 613 377T629 138L631 141Q633 144 637 151T649 170T666 200T690 241T720 295T759 362Q863 546 877 572T892 604Q892 619 873 628T831 637Q817 637 817 647Q817 650 819 660Q823 676 825 679T839 682Q842 682 856 682T895 682T949 681Q1015 681 1034 683Q1048 683 1048 672Q1048 666 1045 655T1038 640T1028 637Q1006 637 988 631T958 617T939 600T927 584L923 578L754 282Q586 -14 585 -15Q579 -22 561 -22Q546 -22 542 -17Q539 -14 523 229T506 480L494 462Q472 425 366 239Q222 -13 220 -15T215 -19Q210 -22 197 -22Q178 -22 176 -15Q176 -12 154 304T131 622Q129 631 121 633T82 637H58Q51 644 51 648Q52 671 64 683H76Q118 680 176 680Q301 680 313 683H323Q329 677 329 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